Fluid Mechanics and Pipe Flow
Pressure, flow, friction, fittings, pumps and meters: the equations of moving fluids, and how to design, size and troubleshoot a piping system, with worked numbers.
Fluid mechanics is the science of liquids and gases at rest and in motion, and pipe flow is its most useful corner: moving water, oil, air and process fluids through pipes, valves and pumps without wasting energy, damaging equipment or starving the user at the end of the line. This guide takes you from the basics (pressure, density, viscosity) through the equations of flow (continuity, Bernoulli, friction, minor losses) to the design of complete systems: pipe sizing, pumps and their curves, flow meters, compressed-air lines and surge. Every worked number is computed, and the same example (water in a DN80 steel line) runs through the pages so the numbers agree. Heat transfer, thermodynamics and HVAC are the other guides in this section; hydraulic and pneumatic actuators are in Motion Control.
Start here: what do you need?
| I need to… | Go to |
|---|---|
| convert a pressure to head, find the force on a wall, or understand viscosity | Fluid Properties and Fluid Statics |
| find a velocity, or understand Bernoulli and the energy equation | Continuity, Bernoulli and the Energy Equation |
| calculate the pressure drop in a pipe | Pipe Friction and the Moody Diagram |
| add up the losses of bends, tees and valves, or size a control valve | Fittings, Valves and Minor Losses |
| choose a pipe size, or work out flow in parallel pipes and loops | Pipe Systems: Series, Parallel and Networks |
| select a pump, read its curve, or stop cavitation | Pumps, Pump Curves and NPSH |
| measure a flow with an orifice, a magnetic meter or a pitot | Flow Measurement |
| size compressed-air pipework or find the cost of a leak | Compressed Air and Gas Flow |
| stop a pipe banging, or diagnose a piping problem | Water Hammer, Cavitation and Troubleshooting |
A suggested order
Read Fluid Properties and Fluid Statics, Continuity, Bernoulli and the Energy Equation and Pipe Friction and the Moody Diagram in order: they carry the ideas that everything else uses. Then Fittings, Valves and Minor Losses and Pipe Systems: Series, Parallel and Networks put a whole system together, Pumps, Pump Curves and NPSH closes the loop with the machine that drives it, and Flow Measurement, Compressed Air and Gas Flow and Water Hammer, Cavitation and Troubleshooting cover measuring, gases and what goes wrong.
- Fluid Properties and Fluid Statics: Density, viscosity and vapor pressure, absolute and gauge pressure, pressure with depth, manometers, forces on walls and buoyancy.
- Continuity, Bernoulli and the Energy Equation: Flow rate and velocity, Reynolds number, laminar and turbulent flow, the Bernoulli equation, the energy equation with a pump and losses.
- Pipe Friction and the Moody Diagram: Darcy–Weisbach, the friction factor, roughness, the Colebrook and Swamee–Jain equations, Hazen–Williams and a fully worked pressure drop.
- Fittings, Valves and Minor Losses: Loss coefficients for bends, tees, entrances and valves, equivalent length, Cv and Kv, and control-valve sizing.
- Pipe Systems: Series, Parallel and Networks: The system curve, pipes in series and parallel, branching networks, Hardy Cross, and choosing a pipe size and velocity.
- Pumps, Pump Curves and NPSH: Pump types, head and efficiency curves, the operating point, affinity laws, cavitation and net positive suction head.
- Flow Measurement: Orifice plates, Venturis, pitot tubes, rotameters, vortex, magnetic, ultrasonic and Coriolis meters, and how to choose.
- Compressed Air and Gas Flow: Compressibility, speed of sound and choked flow, pressure drop in air lines, sizing compressed-air pipework and leaks.
- Water Hammer, Cavitation and Troubleshooting: Pressure surges, how to limit them, cavitation, air in lines, noise and vibration, and a table of common piping problems.
One example, start to finish
Worked example: moving water through a DN80 line
- The job: pump 10 L/s of 20 °C water 12 m up through 100 m of DN80 Schedule 40 steel pipe (inside diameter 77.92 mm) with fittings totaling ΣK = 8.45.
- Speed and regime (Continuity, Bernoulli and the Energy Equation): v = 2.10 m/s and Re = 162 784: turbulent.
- Pipe friction (Pipe Friction and the Moody Diagram): f = 0.0196, so ΔP = 55.1 kPa (5.63 m of head).
- Pump head (energy equation and Fittings, Valves and Minor Losses): 12 m lift + 5.63 m friction + 1.89 m fittings ≈ 19.5 m.
- The real pump (Pumps, Pump Curves and NPSH): a pump with the curve H = 42 − 0.023 Q² would run at 17.7 L/s and 34.8 m on this system, so it needs a valve or a drive to hold 10 L/s, and Water Hammer, Cavitation and Troubleshooting warns what a fast valve closure on this line would do.
Quick reference: the equations
| What you need | Equation |
|---|---|
| Pressure with depth | p = ρ g h |
| Continuity | Q = A v A₁ v₁ = A₂ v₂ |
| Reynolds number | Re = ρ v D ÷ μ |
| Bernoulli | p + ½ ρ v² + ρ g z = constant |
| Pipe friction | ΔP = f (L ÷ D) ρ v² ÷ 2 laminar f = 64 ÷ Re |
| Fittings | ΔP = K ρ v² ÷ 2 |
| Pump power | P = ρ g Q H ÷ η |
| Affinity laws | Q ∝ N, H ∝ N², P ∝ N³ |
| NPSH available | (ps − pv) ÷ ρg + zs − hL |
| Orifice meter | Q = Cd A √(2 ΔP ÷ ρ) ÷ √(1 − β⁴) |
| Water hammer | Δp = ρ a Δv |
Related guides and tools
- Pipe Flow and Pressure Drop Calculator: Darcy–Weisbach with Colebrook, fittings, elevation and a pressure-against-flow chart.
- Pump Power and Sizing Calculator, Compressed Air Consumption Calculator and Cylinder Force and Speed Calculator.
- Psychrometric Calculator: the moist-air side of fluid flow.
- Pneumatic and Hydraulic Actuators and Pressure, Level and Flow Sensors.
- Stress, Strain and Safety Factors: the pipe wall and the pressure it holds; Pressure and Flow Rate converters.
Safety first
- Pressurized pipes and vessels store energy. Depressurize and drain before opening a line, and never go above the rated pressure of the weakest part.
- Water hammer and compressed-air failures are violent. A burst plastic air pipe or a blown hose throws fragments; use only pipe and fittings rated for compressed air, and fit pressure relief where the source can exceed the rating.
- Hot liquids and steam scald. Lag hot pipes and route relief valve discharges to a safe place.
- The numbers here are for study and preliminary design. Pressure equipment, drinking-water systems and anything containing hazardous fluids need a competent engineer, the applicable code (such as ASME B31.3 or local regulations) and tested components.
FAQ
Where do I start if I only want a pipe size?
Read Pipe Friction and the Moody Diagram and the sizing section in Pipe Systems, then use the Pipe Flow and Pressure Drop Calculator. Aim for 1 to 3 m/s for liquids and check the pressure drop.
Why do two sources give different pressure drops?
Mostly roughness and the friction-factor method (Hazen–Williams against Darcy–Weisbach), then fittings. Differences of 10 to 20 percent are normal; use the more conservative one for design.
Is the Darcy friction factor the same as the Fanning factor?
No: Darcy f = 4 × Fanning f. This site always uses the Darcy factor.
Which pages cover compressed air?
Compressed Air and Gas Flow for flow, leaks and pipe sizing, and the Compressed Air Consumption Calculator for demand and compressor size.
Where are heat transfer and steam?
They are the next guides in the Thermal and Fluids section.
References
- White, F. M., Fluid Mechanics, McGraw-Hill; Munson, B. R., et al., Fundamentals of Fluid Mechanics, Wiley; Çengel, Y. A. and Cimbala, J. M., Fluid Mechanics, McGraw-Hill.
- Crane Co., Flow of Fluids Through Valves, Fittings and Pipe, Technical Paper 410; Idelchik, I. E., Handbook of Hydraulic Resistance.
- Hydraulic Institute, Pipe Friction Manual and pump standards (ANSI/HI); Karassik, I. J., et al., Pump Handbook.
- ISO 5167 (differential-pressure flow meters); Wylie, E. B. and Streeter, V. L., Fluid Transients in Systems.
Last reviewed: November 2026. Values are typical; check the standard, the manufacturer’s data and the code that applies to your system.